Greetings. Welcome to the forum.
Reacting first to the title you chose for your question, before I read anything else: No STD causes peeling skin of the hands and feet. If that is your only symptom, you can be sure no STD is responsible.
Now I have read the question itself. My reply still stands. You have been completely evaluated for all STDs that might have caused your mild genital/urinary symptoms.
As for syphilis, very unlikely, regardless of your symptoms. Syphilis is rare in the US at present, only one third as common as HIV, which is rare itself outside selected risk groups; for both infections, almost 75% of new infections are in gay men. Second, if you had syphilis, most likely you would have seen the chancre, as you seem to realize. Third, although it is true that skin rash is the most common symptom of secondary syphilis, in general syphiltic rashes usually do not itch or peel.
That said, syphilis is a protean disease -- i.e. its manifestations are highly varied and atypical cases are common. Given your sexual history, it would be reasonable for you to have a syphilis blood test, but you can expect it to be negative.
I hope this helps. Best wishes-- HHH, MD